# What is the effect on the Berry phase?

• jeon3133

#### jeon3133

No Effort: Member warned that some effort must be shown on homework questions
Homework Statement
Consider a Hamiltonian H[s] that depends on a number of slowly varying parameters collectively called s(t). What is the effect on the Berry phase γn[C] for a given closed curve C, if H[s] is replaced with f[s] H[s], where f[s] is an arbitrary real numerical function of the s?
Relevant Equations
.
Homework Statement :
Consider a Hamiltonian H[s ] that depends on a number of slowly varying parameters collectively called s(t). What is the effect on the Berry phase γn[C] for a given closed curve C, if H[s ] is replaced with f[s ] H[s ], where f[s ] is an arbitrary real numerical function of the s?

Homework Equations :
For any s, we can find a complete orthonormal set of eigenstates Φn of H with eigenvalues En(s):
n = EnΦn
n, Φm) = δnm
.

Attempt at a Solution :
Could you help me to solve this problem?

Do you know the equation for the Berry phase?

In the special case where i and j run over three values,
γn[C] = ∫∫A[C] dA e[s ] ⋅ Vn[s ], ----- (1)
where e[s ] is the unit vector normal to the surface A[C] at the point s, and Vn[s ] is a three-vector in parameter space:
Vn[s ] ≡ i m≠n{(Φn[s ], [∇H [s ]] Φm[s ])* × (Φn[s ], [∇H [s ]] Φm[s ])} × (Em[s ] - En[s ])-2.

I don't understand. Is the closed curve given or is it arbitrary? In the Aharonov-Bohm effect, do you not get different answers if your integration encloses or doesn't enclose the solenoid?

Why do you think that you are told the function is slowly varying?